Question

Tell me everything about Vector

Answer 1

The comprehensive explanation above covers all aspects of vectors.

Answer 2

A vector is a physical quantity that possesses both magnitude and direction, and strictly obeys the laws of vector addition. Quantities that only have magnitude are called scalars.

1. Scalar vs. Vector

• Scalar Quantity: Defined completely by its magnitude only.
  • Examples: Mass, Time, Distance, Speed, Temperature, Energy, Electric Current (though it has direction, it doesn't follow vector addition laws).
• Vector Quantity: Defined by both magnitude and direction, and adheres to vector addition rules.
  • Examples: Displacement, Velocity, Acceleration, Force, Momentum, Torque, Electric Field.

2. Representation of a Vector

• Graphical Representation: A vector is represented by a directed line segment (an arrow). The length of the arrow represents the magnitude of the vector, and the arrowhead indicates its direction.
• Symbolic Representation: A vector is denoted by a bold letter (e.g., $\mathbf{A}$) or a letter with an arrow above it (e.g., $\vec{A}$). Its magnitude is denoted by $|\vec{A}|$ or simply $A$.

3. Types of Vectors

• Unit Vector ($\hat{A}$): A vector with a magnitude of one. It is used solely to specify direction. $\hat{A} = \frac{\vec{A}}{|\vec{A}|}$ Standard unit vectors along X, Y, Z axes are $\hat{i}, \hat{j}, \hat{k}$ respectively.
• Zero Vector (Null Vector, $\vec{0}$): A vector with zero magnitude and an arbitrary direction.
• Position Vector: A vector that specifies the position of a point relative to an origin.
• Displacement Vector: A vector that represents the change in position from an initial point to a final point.
• Equal Vectors: Two vectors are equal if they have the same magnitude and the same direction.
• Negative Vector: A vector having the same magnitude as a given vector but acting in the opposite direction. If $\vec{A}$ is a vector, its negative is $-\vec{A}$.
• Collinear Vectors: Vectors acting along the same line or parallel lines. They can be in the same or opposite directions.
• Coplanar Vectors: Vectors that lie in the same plane.

4. Vector Algebra (Operations)

A. Vector Addition

Vectors can be added graphically or analytically.

• Graphical Methods:

  • Triangle Law of Vector Addition: If two vectors are represented by two sides of a triangle taken in order, their resultant is represented by the third side taken in the opposite order.
  • Parallelogram Law of Vector Addition: If two vectors are represented by the adjacent sides of a parallelogram drawn from a common point, their resultant is represented by the diagonal passing through that common point.
  • Polygon Law of Vector Addition: For adding more than two vectors.

• Analytical Method: If two vectors $\vec{A}$ and $\vec{B}$ make an angle $\theta$ with each other, the magnitude of their resultant $\vec{R} = \vec{A} + \vec{B}$ is: $|\vec{R}| = \sqrt{A^2 + B^2 + 2AB\cos\theta}$ The direction $\alpha$ of the resultant with respect to $\vec{A}$ is: $\tan\alpha = \frac{B\sin\theta}{A+B\cos\theta}$

• Properties of Vector Addition:

  • Commutative Law: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$
  • Associative Law: $(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$

B. Vector Subtraction

Subtracting a vector $\vec{B}$ from $\vec{A}$ is equivalent to adding the negative of $\vec{B}$ to $\vec{A}$: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

C. Multiplication of a Vector by a Scalar

Multiplying a vector $\vec{A}$ by a scalar $k$ results in a new vector $k\vec{A}$. Its magnitude is $k|\vec{A}|$. Its direction is the same as $\vec{A}$ if $k$ is positive, and opposite to $\vec{A}$ if $k$ is negative.

D. Resolution of a Vector into Components

A vector can be resolved into components along perpendicular axes.

• In 2D: A vector $\vec{A}$ in the XY-plane can be written as $\vec{A} = A_x \hat{i} + A_y \hat{j}$, where $A_x = A\cos\theta$ and $A_y = A\sin\theta$ are the scalar components along X and Y axes respectively, and $\theta$ is the angle $\vec{A}$ makes with the positive X-axis. Magnitude: $|\vec{A}| = \sqrt{A_x^2 + A_y^2}$
• In 3D: A vector $\vec{A}$ can be written as $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$. Magnitude: $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$ Direction cosines: $\cos\alpha = \frac{A_x}{|\vec{A}|}$, $\cos\beta = \frac{A_y}{|\vec{A}|}$, $\cos\gamma = \frac{A_z}{|\vec{A}|}$. Note that $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$.

E. Product of Vectors

• Scalar Product (Dot Product): The dot product of two vectors $\vec{A}$ and $\vec{B}$ is a scalar quantity defined as: $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta$ where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$.

  • Properties: Commutative ($\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$), Distributive.
  • In Component Form: If $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ and $\vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k}$, then: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$
  • Applications: Calculating work done ($W = \vec{F} \cdot \vec{d}$), power, finding the angle between vectors, projection of one vector onto another.

• Vector Product (Cross Product): The cross product of two vectors $\vec{A}$ and $\vec{B}$ is a vector quantity defined as: $\vec{A} \times \vec{B} = (|\vec{A}||\vec{B}|\sin\theta)\hat{n}$ where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$, and $\hat{n}$ is a unit vector perpendicular to the plane containing $\vec{A}$ and $\vec{B}$, its direction given by the right-hand rule. The magnitude of $\vec{A} \times \vec{B}$ is the area of the parallelogram formed by $\vec{A}$ and $\vec{B}$.

  • Properties: Anti-commutative ($\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$), Distributive.
  • In Component Form: $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$
  • Applications: Calculating torque ($\vec{\tau} = \vec{r} \times \vec{F}$), angular momentum, magnetic force on a moving charge ($\vec{F} = q(\vec{v} \times \vec{B})$), finding a vector perpendicular to two given vectors, area of a triangle or parallelogram.

5. Applications of Vectors

Vectors are fundamental in various fields:

• Physics: Describing motion (displacement, velocity, acceleration), forces, momentum, fields (electric, magnetic, gravitational), fluid dynamics, waves.
• Mathematics: Geometry (lines, planes, 3D shapes), linear algebra, calculus (vector calculus).
• Engineering: Structural analysis, robotics, computer graphics, navigation systems, aerospace.